Bargaining with habit formation - Duke...

Econ Theory (2017) 64:477–508DOI 10.1007/s00199-016-0994-z

RESEARCH ARTICLE

Bargaining with habit formation

Bahar Leventoglu1

Received: 31 January 2014 / Accepted: 13 July 2016 / Published online: 28 July 2016© Springer-Verlag Berlin Heidelberg 2016

Abstract Habit formation is a well-documented behavioral regularity in psychologyand economics; however, its implications on bargaining outcomes have so far beenoverlooked. I study an otherwise standard Rubinstein bargaining model with habit-forming players. In equilibrium, a player can strategically exploit his opponent’s habit-forming behavior via unilateral transfers off the equilibrium path to generate endoge-nous costs and gain bargaining leverage at no cost to himself on the equilibrium path.Uncertainty about habit formation may lead to delay in agreement.

Keywords Bargaining · Habit formation · Behavioral economics · Uncertainty

JEL Classification C72 · C78

1 Introduction

Habit formation is a well-documented behavioral regularity in psychology and behav-ioral economics (Camerer and Loewenstein 2004). Accordingly, human beings formhabits for consumption and their current satisfaction level tends to be highly correlatedwith their past consumption level.Habit formation has been incorporated extensively toa number of research programs in economics, finance, international conflict, manage-

Thanks go to Alexandre Debs, Robert Powell, Huseyin Yildirim and two anonymous referees for helpfulcomments and suggestions.

B Bahar [email protected]

1 Department of Political Science, Duke University, 140 Science Drive, Durham, NC 27708, USA

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s00199-016-0994-z&domain=pdf

478 B. Leventoglu

ment science and social psychology among others.1 However, the formal bargainingliterature has so far overlooked its implications in bargaining.

Habitual behavior may be induced in everyday bargaining situations, e.g., betweena drug dealer and his client, or may arise in economic and political interactions suchas an ongoing dispute between a sanctioning country and its target. For example, inhis September 27, 2009 New York Times article on the US–Iran relations in regardto nuclear non-proliferation, Roger Cohen argues that “sanctions won’t work” partlybecause “Iran is inured to sanctions after years of living with them.” In this paper,I examine the role of habit formation in bargaining and, in particular, explore howplayers can exploit adversary’s habit-forming behavior to generate bargaining leveragein negotiations.

I introduce habit formation into an otherwise standard bargainingmodel (Rubinstein1982).2 Two players collectively receive a flow of v units of consumption good over aninfinite time horizon. The players can consume the good only if they mutually agree toshare it. The players discount their future payoffs. They make offers in an alternatingfashion in the beginning of every period. If a player’s proposal is accepted, the twoplayers share v accordingly forever. If a proposal is rejected, v of the current periodperishes. I extend this standard model as follows: When an offer is rejected, player 1can make a unilateral transfer using his other resources to player 2. In the benchmarkcase, I model player 2 as an individual that forms habits for consumption and hercurrent satisfaction levels tend to be highly correlatedwith her past consumption levels(Camerer and Loewenstein 2004). In particular, player 2 uses her consumption fromthe previous period as a reference point and pays a cost if her current consumption fallsbelow her past consumption level (Rozen 2010).3 Her cost is a linear function of thedifference between current and previous consumption levels. I refer to the coefficientof this linear function as the cost coefficient or marginal cost.

In a subgame perfect equilibrium, player 1 exploits player 2’s habit- formingbehavior to generate endogenous costs for player 2. Specifically, as long as offersare rejected, which happens only off the equilibrium path, player 1 alternates betweenmaking a unilateral transfer one period and no transfer the following period. Sinceplayer 2 pays a cost only when she consumes less than what she has consumed in theprevious period, the alternating scheme of unilateral transfers lowers player 2’s con-tinuation payoff off the equilibrium path when an offer is rejected. In turn, she acceptslower offers on the equilibrium path. This bargaining leverage comes to player 1 atno cost since he exploits player 2’s habit formation by unilateral transfers only off

1 For example, habit formation has been used to account for the consumption data in the USA as wellas other countries (Ferson and Constantinides 1991; Braun et al. 1993) and some notable asset pricinganomalies such as the equity premium puzzle (Abel 1990; Constantinides 1990; Campbell and Cochrane1999). Scholars in growth economics, for example, Carroll et al. (1997), and monetary economics, forexample Fuhrer (2000), have also utilized rational models of habit formation for their explanatory andpredictive power.2 See Osborne and Rubinstein (1990) for an extensive review of the literature on bargaining models.3 Also see Apesteguia and Ballester (2009). The standard bargaining model can be extended by alternativebehavioral assumptions, such as endowment effects. See Kahneman et al. (1991) for a discussion of suchanomalies.

123

Bargaining with habit formation 479

the equilibrium path without actually making any transfer on the equilibrium path. Inaddition, a higher cost coefficient benefits player 1.

The benchmark model applies to situations where player 1 represents an econom-ically powerful player, for whom the unilateral transfer is not large enough to causehabitual behavior. However, such transfers may have domestic political consequencesthat induce habitual behavior by the player. For example, in case of an internationalconflict, deviating from the status quo by increasing the level of unilateral transferto an adversary may create political costs for the domestic government. Under suchcircumstances, player 1 continues to exploit the other player’s habitual behavior albeitto a lesser extent.

Player 2 may enjoy a bump in her utility from excess consumption in comparisonwith her past consumption. One may expect more leverage for player 1, since nowhe may try to exploit this bump in the first period as well. However, such habitualbehavior improves player 2’s equilibrium share. This counterintuitive result stemsfrom the observation that, when player 2 rejects an offer off the equilibrium path andher offer is accepted the next period, she enjoys a bump, which improves her payofffrom rejecting an offer and therefore player 1’s offer.

The model predictions hold when player 2 can refuse a unilateral transfer. Theyalso survive in the presence of asymmetric information; however, an equilibrium delaymay emerge.When player 1 is uncertain about whether player 2 exhibits habit-formingbehavior or not, if player 2 makes the first offer and she is not a habit-forming type,she can credibly signal her type to player 1 in a separating equilibrium by facing arisk of delay in negotiations. Such a separating equilibrium is possible only if themarginal cost from habits is not too high for a habit-forming type. Otherwise, bothtypes pool bymaking the offer that the habit-forming type wouldmake in the completeinformation game. A higher marginal cost coefficient benefits player 1 by increasinghis expected payoff. It also increases the likelihood of a delay, which only harmsnon-habit-forming player 2. The potential delay disappears when player 1 makes thefirst offer. More importantly, player 1 continues to exploit player 2’s habit formationwithout actuallymaking any transfers in the equilibrium of the incomplete informationgame. That is, the threat of the unilateral transfer is sufficient to generate bargainingleverage for player 1.

Closest to my work is the literature on bargaining games in which players canendogenously determine disagreement payoffs (Haller and Holden 1990; Fernandezand Glazer 1991; Avery and Zemsky 1994; Busch and Wen 1995). In the standardRubinsteinmodel, disagreement payoffs are fixed and there is a unique perfect equilib-riumwhich is efficient and has stationary equilibrium offers. In contrast, disagreementpayoffs are endogenous in the aforementioned articles, and there may exist multipleequilibria, some of which may be inefficient, explaining wasteful phenomena such asstrikes, delay in bargaining and wars. The driving force behind these results is thatactions that are taken in the stage game that determines the disagreement payoffs maydetermine equilibria that are going to be played in future. This allows for multipleequilibria, some of which are inefficient.

My model is similar to these earlier works in the sense of disagreement payoffsbeing endogenous. However, in contrast, a player’s past consumption level is a payoff-relevant state of the game in my model. This has nontrivial implications on model

123

480 B. Leventoglu

predictions that differ from those of the aforementioned work. For example, Hallerand Holden (1990) and Fernandez and Glazer (1991) study wage bargaining betweena firm and a union. In their model, the union may strike in case of disagreement, whichis costly for both the firm and the union workers. Not striking is a Nash equilibriumof the stage game. Therefore, the unique equilibrium of the standard Rubinstein gamewithout a strike option is an equilibrium of the game with the strike option (Busch andWen 1995, Corollary 1).

Likewise, inmymodel, player 1 does notmake any unilateral transfers in the uniqueNash equilibrium of the stage game. However, Corollary 1 of Busch and Wen (1995)does not apply. In particular, whenmarginal cost from habits exceeds a threshold, thereis no equilibrium in my model in which player 1 does not make any unilateral transferon and off the equilibrium path. Therefore, the unique equilibrium of the standardRubinstein game without a transfer option is not an equilibrium in my model. Thisis because a transfer today changes preferences over consumption in future via habitformation and player 1 finds it optimal to manipulate player 2’s future preferences.This is not solely an equilibrium phenomenon as in the earlier work, in which actionstoday do not change fundamentals in future. In these earlier models, players continueto have the same preferences regardless of what actions have been played in the past;however, players’ past actions may determine equilibria that are going to be played infuture.

Finally, the prospect theory of international conflict (Levy 1996, 1997a, b also seeBerejikian 2004) hypothesizes that political leaders of adversary states behave differ-ently when they are bargaining over gains than when they are bargaining over losses(Levy 1996). Although this is close to my work in terms of exploring nonstandardpreference patterns in bargaining, my work does not rest on prospect theory: First,players’ payoff functions are weakly concave everywhere in my model,4 and second,the insights that are derived frommymodel are fundamentally different than that comefrom prospect theory.

I introduce the complete information benchmark model in the next section and dis-cuss the equilibrium in Sect. 3. I consider three extensions of the benchmark modelin Sect. 4, in which both players exhibit habit-forming behavior, habit-forming playeralso enjoys a bump in utility from excess consumption, and player 2 can refuse uni-lateral transfers. I also study a case in which player 1 is uncertain whether player 2exhibits habit-forming behavior or not. I defer all the technical analysis to the appendix.

2 Bargaining with habit-forming players

The benchmark model is an extension of the Rubinstein (1982) bargaining model:There are two players 1 (he) and 2 (she). The players bargain over a flow of v units ofa perishable consumption good that they can share and consume only if they mutuallyagree to do so. The players make offers in an alternating and deterministic order. The

4 Prospect theory (Kahneman and Tversky 1979) postulates that an individual evaluates alternatives withrespect to a reference point and assigns value to gains and losses with respect to the reference rather thanto final assets. The value function is generally concave for gains, convex for losses and steeper for lossesthan for gains.

123


player that is selected to make an offer makes a proposal x for player 2’s share andtherefore v − x for player 1’s share. If an agreement is reached, the players consumetheir shares of the flow thereafter. When a player rejects a proposal, he makes the nextproposal. However, unlike in theRubinsteinmodel, player 1makes a unilateral transferof y ≥ 0 to player 2 before the next proposal, which costs player 1 ψy, ψ ≥ 0. Player2 consumes y and the game proceeds to the next period and continues until one playeraccepts the other’s proposal. The assumption that player 2 consumes the unilateraltransfer without refusing it makes my model closer to the models on bargaining withendogenously determined disagreement payoffs (Haller and Holden 1990; Fernandezand Glazer 1991; Avery and Zemsky 1994; Busch and Wen 1995). In these models,a player can take a unilateral action that changes players’ disagreement payoffs. InSect. 4.3, I also consider a case in which player 2 can reject unilateral transfers.

Player i discounts future payoffs by δi ∈ [0, 1). Let zi,t denote player i’s consump-tion in period t, which also includes the cost of unilateral transfer made in the period.Player 1’s lifetime utility is given by

∑

t

δt1z1,t

Player 2 exhibits habit-forming behavior. Her lifetime utility is given by

∑

t

δt2

(z2,t − φ[z2,t−1 − z2,t ]+

)

where

[z2,t−1 − z2,t ]+ ={

z2,t−1 − z2,t if z2,t−1 > z2,t0 otherwise

φ[z2,t−1 − z2,t ]+ captures player 2’s cost for her habit from past consumption. Ifplayer 2’s current consumption is at least as much as her consumption in previousperiod, i.e., z2,t ≥ z2,t−1, then player 2 does not bear any additional cost, and herper-period payoff is her current consumption. If her current consumption is less thanher consumption in the previous period, i.e., z2,t < z2,t−1, then player 2 pays a cost ofφ(z2,t−1 − z2,t ) and her per-period payoff is her current consumption minus the cost.This cost increases linearly with the difference between consumption levels in currentand previous periods. Themarginal cost coefficient φ ≥ 0measures how costly it is forthe player when there is a gap between the current and past consumption levels. Thehigher φ is, the greater costs player 2 pays. The only part of the cost that is exogenousis φ.

The game starts in period t = 1 with z20 = 0. The benchmark model capturesscenarios in which player 1’s income from other resources is large enough in compar-ison with unilateral transfers he could make so that he does not suffer any cost fromhabit-forming behavior.

123

482 B. Leventoglu

Next I studyMarkov perfect equilibrium inwhich offers are accepted immediately.5

3 Equilibrium

An equilibrium is Markov perfect if strategies depend only on payoff- relevant stateof the game. Since player 2’s previous period consumption is the only payoff-relevantstate in any period, I define player strategies as function of player 2’s previous periodconsumption and drop the time index below.

Let z denote player 2’s previous period consumption level. xi (z) ∈ [0, v] is playeri’s offer for player 2’s share; yi (z) ≥ 0 is player 1’s unilateral transfer when player i’soffer is rejected; ai (x; z) ∈ {accept, reject} is player i’s decision to accept or rejectan offer of x made by the other player. xi (z) can be set arbitrarily when it is not i’sturn to make an offer and ai (x; z) can be set arbitrarily when it is i’s turn to makean offer. Denote player 1’s strategy by σ1(z) = (x1(z), y1(z), y2(z), a1(x; z))t=1,2,...,

and player 2’s by σ2(z) = (x2(z), a2(x; z))t=1,2,.... A strategy profile (σ1, σ2) is asubgame perfect equilibrium if the continuation of (σ1, σ2) forms a Nash equilibriumat every subgame. The game starts with z = 0 and player 1 makes the first offer.Otherwise, z is determined by player 1’s previous period unilateral transfer.

When there is no habit formation, the model reduces to the standard Rubinsteinbargaining game with a unique and efficient equilibrium with stationary equilibriumoffers of

x∗1 = δ2(1 − δ1)

1 − δ1δ2v and x∗

2 = 1 − δ1

1 − δ1δ2v

which are accepted immediately.Let

φ∗ = (1 − δ1)ψ

δ1(1 − δ2).

Before characterizing the equilibrium, I will give an intermediate result to differentiatemy work from the aforementioned work. I defer all the proofs to the “Appendix.”

Proposition 1 If φ > φ∗, then y1(z) = y2(z) = 0 for all z cannot hold in equilib-rium.

This result implies that a positive unilateral transfer is part of player 1’s equilibriumstrategy when φ > φ∗. This contrasts with earlier work that studies bargaining withendogenously determined disagreement payoffs (Haller and Holden 1990; Fernandezand Glazer 1991; Avery and Zemsky 1994; Busch andWen 1995). Like in these earliermodels, disagreement payoffs are endogenous in my model. However, unlike them,player 2’s past consumption level is a payoff-relevant state of the game in my model.This has nontrivial implications for model predictions. The Nash equilibrium of thestage game of my model involves no unilateral transfers, but a positive unilateral

5 There may exist other types of subgame perfect equilibrium, which I do not study here.

123


transfer becomes part of player 1’s equilibrium strategy when φ > φ∗ of the game.Therefore, the unique equilibrium of the standard Rubinstein game without a transferoption is not an equilibrium when φ > φ∗ in my model even though no transfer ispart of the unique Nash equilibrium of the stage game. In other words, Corollary 1 ofBusch and Wen (1995) does not apply.

In the rest, I focus onMarkov perfect equilibria in which offers are accepted imme-diately. Let

z∗ = x∗1

1 + (1 − δ2)φ, x2 = (1 + (1 − δ2)φ)(1 − δ1)v

1 − δ1δ2 + (1 − δ2)φ − δ2(1 − δ1)ψ, and

z = δ2 x21 + (1 − δ2)φ

The following theorem summarizes the Markov perfect equilibrium of the gamewith habit formation at which equilibrium offers are accepted immediately.

Theorem 2 (i) (Low Cost) If φ ≤ φ∗, the following is a Markov perfect equilibrium:Player 1 does not make any unilateral transfer when an offer is rejected, i.e.,y1(z) = y2(z) = 0 for all z; if player 1 made a unilateral transfer of z afterrejecting 2’s offer in the previous period, then player 1 offers

x1(z) ={

z∗ + (1 − δ2)φ(z∗ − z) if z ≤ z∗z∗ if z ≥ z∗

he accepts any offer smaller than or equal to x∗2 and rejects any other offer; given

that player 2’s past period consumption is z, player 2 offers x∗2 , i.e., x2(z) = x∗

2for all z, accepts any offer greater than or equal to x1(z) and rejects any otheroffer.

(ii) (High Cost) If φ > φ∗, the following is a Markov perfect equilibrium: Player 1does not make any unilateral transfer when 2 rejects his offer and he makes aunilateral transfer of z when he rejects 2’s offer, i.e., y1(z) = 0 and y2(z) = zfor all z; if player 1 made a unilateral transfer of z after rejecting 2’s offer in theprevious period, then player 1 offers

x1(z) ={

z + (1 − δ2)φ(z − z) if z ≤ zz if z ≥ z

he accepts any offer smaller than or equal to x2 and rejects any other offer; giventhat player 2’s past period consumption is z, player 2 offers x2, i.e., x2(z) = x2for all z, accepts any offer greater than or equal to x1(z) and rejects any otheroffer.

When φ ≤ φ∗, there is no unilateral transfer on the equilibrium path and x1(0) =x∗1 . That is, the players offer x∗

1 and x∗2 in equilibrium, so habit formation does not

have any effect on the equilibrium outcome.When φ > φ∗, y2(z) = z and x1(0) = δ2 x2 < δ2x∗

2 = x∗1 since x2 < x∗

2 . Thatis, player 1 takes advantage of player 2’s habit formation to decrease the equilibrium

123

484 B. Leventoglu

offers for player 2. This bargaining leverage comes to player 1 at no additional costsince offers are accepted immediately in equilibrium and player 1 never makes aunilateral transfer on the equilibrium path.

The timing of 1’s unilateral transfer off the equilibrium path reveals how player 1takes advantage of player 2’s habit-forming behavior. Player 1 can make a unilateraltransfer both after rejecting player 2’s offer and after player 2 rejects his offer. It isoptimal for him to alternate between making a unilateral transfer one period and notransfer the following period off the equilibrium path. This creates a cost for player 2in periods with no unilateral transfer and decreases her continuation payoff when anoffer is rejected off the equilibrium path. Along this alternating path, Player 1 makesthe unilateral transfer after he rejects player 2’s offer. Intuitively, player 2 collects alarger share of surplus when her offer is accepted. Player 1 can recover some of thissurplus by threatening player 2 with a rejection. After rejecting player 2’s offer, hecan unilaterally transfer the amount that he will propose next period. This transfer willcreate cost for player 2 in the next period if she rejects it; therefore, she will acceptsmaller offers next period. In turn, player 1’s continuation payoff will be higher, sincehe can induce player 2 to accept smaller offers next period. Anticipating player 1’sactions off the equilibrium path and his continuation payoff, player 2 rationally makesand accepts smaller offers in comparison with the case without habit formation.

The closed-form solution yields intuitive comparative statics. dxidφ < 0 and dxi

dψ > 0for both i = 1, 2whenφ > φ∗. In other words, both players decrease their equilibriumoffers for player 2’s sharewhen themarginal cost coefficientφ increases or themarginal

cost of unilateral transfer,ψ, decreases. Also dφ∗dψ > 0 and ∂2xi

∂ψ∂φ> 0 for both i = 1, 2

when φ > φ∗. In words, as player 1’s cost of unilateral transfers gets larger, it becomesmore difficult for him to exploit player 2’s habit-forming behavior.

This completes the discussion of the benchmark model. Next, I discuss the exten-sions of the model.

4 Discussion and extensions

If unilateral transfer constitutes a small fraction of player 1’smomentary consumption,it may not induce habitual behavior. Therefore, the benchmark model can be thoughtof capturing an interaction between an economically powerful player (player 1) and asmall player by assuming away habitual behavior by player 1. I extend this model inseveral ways below. First, I study bargaining under habitual behavior by both players.Next, I extend the benchmark model by assuming that player 2 may also enjoy a bumpin utility from excess consumption in comparison with her past consumption. Then,I study the case in which player 2 can refuse unilateral transfers. Finally, I study anincomplete information model in which player 1 is uncertain about whether player 2exhibits habit-forming behavior or not.

4.1 Two-sided habit formation

I continue assuming that only player 1 canmakeunilateral transfers as in the benchmarkmodel. However, assume that player 1 also pays a cost due to habit formationwhenever

123


his consumption level falls below his consumption in the previous period. That is, ifhe consumes z1,t , his lifetime utility is given by

∑

t

δt1

[z1,t − γ [z1,t−1 − z1,t ]+

]

where

[z1,t−1 − z1,t ]+ ={

z1,t−1 − z1,t if z1,t−1 > z1,t0 otherwise

γ ≥ 0, and γ [z1,t−1− z1,t ]+ captures player 1’s cost from his habit for past consump-tion.

The statement of Theorem 2 continues to hold in this extended model after redefin-ing the values of φ∗, z and x2 as follows:

φ∗ = (1 − δ1)(1 + γ )ψ

δ1(1 − δ2),

x2 = (1 + (1 − δ2)φ)(1 − δ1)v

1 − δ1δ2 + (1 − δ2)φ − δ2(1 − δ1)(1 + γ )ψ, and z = δ2 x2

1 + (1 − δ2)φ

The following summarizes the equilibrium:

Theorem 3 The statement of Theorem 2 with redefined values of φ∗, z and x2 holds.

The newφ∗ is larger, so player 1 can exploit player 2’s habitual behavior in a smallerparameter region in comparisonwith the benchmark. Also player 2’s equilibrium shareis higher. This is due to increased cost of unilateral transfers in this new setting. Thecost has two components: The direct cost of y is the decrease in his consumption byψy as in the benchmark model, and γ y is the cost thanks to habit formation.

If player 2 were to make the first offer in the standard Rubinstein model, she wouldhave offered x∗

2 . So a first mover advantage in the standard model is evident fromx∗1 = δ2x∗

2 . That is, when player 1 makes the offer, he proposes x∗1 , which is less

than x∗2 , the amount player 2 would offer if she made the first offer. In the model with

two-sided habit formation of this section, player 2’s share increases with γ. That is,player 1 continues to exploit player 2’s habitual behavior to a smaller extent. One maybe tempted to think that player 1’s bargaining leverage is due to first mover advantagesince he loses his leverage as he exhibits habitual behavior. However, x1(0) = δ2 x2 asin the standard Rubinstein model, that is, a first mover advantage is present but alsox2 < x∗

2 when φ > φ∗, so that player 1 can exploit player 2’s habitual behavior evenwhen player 2 makes the first offer.

4.2 Joy from excess consumption

The benchmark model assumes only negative utility when current consumption dropspast period consumption. Equally plausible, an agent may enjoy a bump in utility if

123

486 B. Leventoglu

she consumes more than the previous period (Orrego 2014). In order to capture thisscenario, modify the benchmark model as follows: Let λ ≥ 0 be the marginal utilitythat player 2 derives from consuming additional amount in comparison with her pastperiod consumption. Her lifetime utility is given by

∑

t

δt2

(z2,t − φ[z2,t−1 − z2,t ]+ + λ[z2,t − z2,t−1]+

)

where φ[z2,t−1 − z2,t ]+ is the negative utility from falling below past consumptionlevel as before and λ[z2,t − z2,t−1]+ is the positive utility bump from consuming morethan the past consumption.

Redefine the following variables:

φ∗ = (1 + (1 − δ2)λ)(1 − δ1)ψ

δ1(1 − δ2),

z∗ = 1 + (1 − δ2)λ

1 + (1 − δ2)φx∗1

x2 = (1 + (1 − δ2)φ)(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)(φ − δ1δ2λ) − δ2(1 − δ1)ψ(1 + (1 − δ2)λ)

and

z = 1 + (1 − δ2)λ

1 + (1 − δ2)φδ2 x2

Then the following summarizes the equilibrium:

Theorem 4 (i) (Low Cost) If φ ≤ φ∗, the following is a Markov perfect equilibrium:Player 1 does not make any unilateral transfer when an offer is rejected, i.e.,y1(z) = y2(z) = 0 for all z; if player 1 made a unilateral transfer of z afterrejecting 2’s offer in the previous period, then player 1 offers

x1(z) ={

x∗1 − (1 − δ2)φz if z ≤ z∗

z∗ if z ≥ z∗

he accepts any offer smaller than or equal to x∗2 and rejects any other offer; given

that player 2’s past period consumption is z, player 2 offers x∗2 , i.e., x2(z) = x∗

2for all z, accepts any offer greater than or equal to x1(z) and rejects any otheroffer.

(ii) (High Cost) If φ > φ∗, the following is a Markov perfect equilibrium: Player 1does not make any unilateral transfer when 2 rejects his offer and he makes aunilateral transfer of z when he rejects 2’s offer, i.e., y1(z) = 0 and y2(z) = zfor all z; if player 1 made a unilateral transfer of z after rejecting 2’s offer in theprevious period, then player 1 offers

x1(z) ={

δ2 x2 − (1 − δ2)φz if z ≤ zz if z ≥ z

123


he accepts any offer smaller than or equal to x2 and rejects any other offer; giventhat player 2’s past period consumption is z, player 2 offers x2, i.e., x2(z) = x2for all z, accepts any offer greater than or equal to x1(z) and rejects any otheroffer.

The new φ∗ is larger in this case, too. So player 1 can exploit player 2’s habitualbehavior in a smaller parameter region in comparison with the benchmark. However,x1(0) and x2 are both larger in comparison with the benchmark; therefore, player 2’sequilibrium share is higher.When player 2 rejects an offer off the equilibrium, the nextoffer is accepted, fromwhich she collects a bump in utility. Therefore, her continuationpayoff from rejecting an offer increases, improving equilibrium offers for her.

4.3 Mutual consent for unilateral transfers

The benchmark model accounts for scenarios in which player 2 cannot refuse toconsume unilateral transfers. For example, during trade sanctions, a sanctioning state(player 1) bans exports to a target state (player 2). In this case, a unilateral transfermay represent temporary sanctions relief. Even if the political elite in the sanctionedstate (target) may prefer to reject such unilateral transfers for strategic reasons, goodsfrom the sanctioning state may find their way to the target through indirect routeswhen sanctions are lifted temporarily even if imports from the sanctioning country arebanned in the target state.

The assumption of inability to refuse unilateral transfers would be violated if thetarget state can control its borders. Likewise, when a drug dealer (player 1) offers somefree drug (unilateral transfer), his client (player 2) has the option of refusing it despiteher addiction to the drug. In order to capture such scenarios, I assume in this section thatplayer 2 can accept or reject a unilateral transfer.When she rejects a unilateral transfer,she does not consume anything that period.6 Let Ti (y; z) ∈ {accept, reject} is player2’s decision to accept or reject a unilateral transfer of y made by player 1 after playeri’s offer is rejected and incorporate Ti (y; z) into her strategy. In the “Appendix,” Ishow that Theorem 2 holds under this relaxation when player 2 is impatient enough.7

The intuition behind this result is as follows. Player 1 makes no unilateral transferwhen player 2 rejects his offer, i.e., y1(z) = 0, whether player 2 can or cannot refuseunilateral transfers. Similarly, y2(z) = 0 when φ ≤ φ∗ in the benchmark model andplayer 2’s ability to refuse y2(z) = 0 is inconsequential so that y2(z) = 0whenφ ≤ φ∗in the new model as well. Therefore, the same equilibrium obtains when φ ≤ φ∗. Inthe case of φ ≥ φ∗, if player 2 is not too patient, she always accepts the unilateraltransfer of Theorem 2 when equilibrium offers are also as in Theorem 2. Therefore,player 2’s ability to refuse does not change the equilibrium outcome in this case either.

6 See Ponsatí and Sákovics (1998) for bargaining with outside options.7 More formally, I show that Theorem 2 holds when δ2(1 + (1 − δ2)φ) ≤ 1. This sufficiency conditionsimplifies the exposition of the proof. The same result may also follow for larger values of δ2, and I do notexplore the necessary condition in this extension.

123

488 B. Leventoglu

4.4 Incomplete information

This section studies consequences of habit-forming behavior under incomplete infor-mation8. Suppose that player 1 does not knowwhether player 2 exhibits habit formationor not. Let φ ∈ {φl = 0, φh}, φ∗ < φh . If φ = φl = 0, then player 2 does not exhibithabit formation. If φ = φh, then player 2 exhibits habit formation. Player 2’s typeis her private information. It is common knowledge that she exhibits habit formationwith probability θ, i.e., Pr(φ = φh) = θ . The outcome is predicted by Bayesian Nashequilibrium.

Under complete information, if φ = φl , then yi = 0 for i = 1, 2 and theequilibrium offers are given by

x∗1 = δ2(1 − δ1)


2 = 1 − δ1

1 − δ1δ2v

and if φ = φh, the equilibrium outcomes are given by

x1(0) = xh1 = (1 − δ1)δ2v

(1 − δ2)(1 + φh)and x2(z) = xh

2 = 1 + (1 − δ2)φh

(1 − δ2)(1 + φh)(1 − δ1)v,

in that case, y1(0) = 0 and y2(0) = xh1 .

Assume that player 2 makes the first offer. When different types of player 2 makedifferent offers in a separating equilibrium, the continuation game reduces to a com-plete information game. Define

φ = δ2(1 − δ1δ2)

(1 − δ2)(δ22 − 1 + δ1δ2)

− 1 > φ∗ and α = x∗2 − xh

2

x∗2 − δ2x∗

1

The following proposition summarizes the Bayesian equilibrium when player 2makes the first offer.

Proposition 5 Assume that player 2 makes the first offer.

(i) If δ1 <1−δ22δ2

or φh ∈ [φ∗, φ], the following is a separating equilibrium: Type φl

offers x∗2 , which player 1 accepts with probability 1−α. Type φh offers xh

2 < x∗2 ,

which player 1 accepts with probability 1. Player 2 rejects any other offer biggerthan xh

2 , and accepts any other offer smaller than xh2 . If player 2 offers x∗

2 , thenplayer 1 updates his belief to Pr(φ = φh) = 0, otherwise, he updates his belief toPr(φ = φh) = 1 and the players play the equilibrium of the associated completeinformation games. The equilibrium payoff of both types of player 2 is xh

2 .

(ii) If δ1 >1−δ22δ2

and φh > φ, there does not exist any separating equilibrium. The

following is a pooling equilibrium: Both types of player 2 offer xh2 . Player 1 rejects

offers x > xh2 and accepts offers x ≤ xh

2 . If player 2 offers x �= xh2 , then player 1

8 e.g., see Muthoo (1994).

123


updates his belief to Pr(φ = φh) = 1 and plays according to the equilibrium ofthe associated complete information game. The equilibrium payoff of both typesof player 2 is xh

2 .

Since xh2 is a decreasing function of φh, α increases with φh . That is, the likelihood

of delay in agreement when player 1 receives an offer of x∗2 increases. However, since

player 1 is indifferent between accepting and rejecting x∗2 , this increase in the likeli-

hood of delay does not decrease his expected payoff. In contrast, since xh2 decreases

with φh , his payoff of v − xh2 increases. Type φl ’s expected payoff decreases because

of the delay.9

Now consider the game in which player 1 makes the first offer. Suppose that theplayers play the equilibrium in Proposition (5) in a continuation game when player2 rejects player 1’s offer. Then there is no separating equilibrium in which differenttypes of player 2 separate themselves by their acceptance/rejection decision. This isbecause player 2 of type φh can achieve a higher payoff by imitating φl , after whichthe game turns into a complete information game with φ = φl and φh collects thehighest payoff she can. Therefore, the continuation game after a potential rejection isan incomplete information game inwhich player 2makes the offer. By Proposition (5),both types of player 2 are xh

2 whether they play a separating or pooling equilibriumin the continuation incomplete information game. Then, the payoff from rejectingplayer 1’s offer is δ2xh

2 for both types of player 2, so that player 1 offers δ2xh2 and

both types accept in equilibrium. The following proposition summarizes a poolingBayesian equilibrium when player 1 makes the first offer.

Proposition 6 Assume that player 1 makes the first offer. Player 1 offers δ2xh2 . Both

types of player 2 accept offers x ≥ δ2xh2 and reject offers x < δ2xh

2 . If an offer isrejected, player 1 updates his belief to Pr(φ = φh) = 1, and the players play theequilibrium strategies of the complete information game with φ = φh .

If player 1 had known that φ = φh, he would have offered xh1 < δ2xh

2 . Thus, player2 of type φh collects an information rent of R(φh) = δ2xh

2 − xh1 . In the complete

information game, player 2 of type φh is hurt by an increase in φh, since xh1 and xh

2are decreasing functions of φh . She is also hurt by an increase in φh in the incom-plete information game for the same reason. However, both R(φh) and R(φh)/xh

1 areincreasing functions of φh . That is, player 1 pays her a larger information rent in bothabsolute and relative values as φh increases.

5 Conclusion

In this paper, I examine the role of habit formation in bargaining and show how a playercan exploit his opponent’s habit-forming behavior to generate endogenous costs forthe opponent. Habit- forming behavior brings in new strategic tools and incentiveswhich can be utilized off the equilibrium of a complete information game in orderto leverage an agent’s bargaining power over the habit- forming opponent at no cost

9 See Merlo and Wilson (1998) for efficient delays in bargaining.

123

490 B. Leventoglu

to the agent. The qualitative features of the models are robust to several extensionsalbeit to a differing extent. Introduction of informational asymmetry may cause delayin negotiations without changing the qualitative predictions.

Among other economic and political areas, these findings have direct implicationsin international bargaining. In particular, the carrot–stick approach is a preferred policyin international negotiations. The conventional wisdom is that offering a combinationof rewards and punishments may be effective in getting one’s opponent to make con-cessions in bargaining. My findings suggest that, even a “free carrot” may sometimesimprove the hand of a negotiating country if, for example, such free carrots createpolitical costs in the adversary state via citizens’ habit-forming behavior.

The international relations literaturemostly studies such problems in isolation, inde-pendent of other possible coercive tools, such as the actual use of force, for example,see Jonge Oudraat (2000) for a criticism of the sanctions literature. It is quite possiblethat such policies can work more effectively when they are a part of a comprehensivestrategy. That includes the use of force as an outside option. I leave this problem forfuture research.

Appendix: Preliminaries

In equilibrium, each player chooses an optimal action at every subgame. I will assumethat a player accepts an offer when he is indifferent between accepting and rejectingit given the continuation equilibrium. If player j is better off by accepting player i’soffer, then player i can increase his/her payoff by slightly decreasing player j’s share,which player j will continue to accept. Therefore, player i’s optimal offer makesplayer j indifferent between accepting and rejecting i’s offer. Consider an equilibriumat which offers are accepted. Consider an off equilibrium path in period t . Let z denoteplayer 2’s consumption in the previous period on the given off-the-equilibrium path.The conditions for the equilibrium offers that are accepted are given as follows. Ifplayer 1 makes the offer in period t, then

x1(z)

1 − δ2− φ [z − x1(z)]+ = (y1(z) − φ [z − y1(z)]+)

+ δ2

(x2(y1(z))

1 − δ2− φ [y1(z) − x2(y1(z))]+

)(1)

where player 1’s equilibrium strategy is to offer x1(z) given that player 2 consumedz in the previous period. The left-hand side of this equation is player 2’s payoff ifshe accepts player 1’s offer. x1(z)

1−δ2is player 2’s lifetime utility from consuming x1(z)

forever, and −φ [z − x1(z)]+ is the cost that she pays in period t. Since there is nochange in the consumption level thereafter, there is no further cost after period t.The right-hand side is player 2’s payoff if she rejects the offer x1(z). In that case,player 1 makes a unilateral offer of y1(z), which player 2 consumes and pays thecost of φ [z − y1(z)]+ in period t, and then her equilibrium offer of x2(y1(z)) willbe accepted in period t+ 1, which she will consume thereafter, and she will pay aone-period cost of −φ [y1(z) − x2(y1(z))]+ in period t + 1.

123


Equality (1) is necessary and sufficient for x1(z) to be part of an equilibrium givenx2(), y1() and y2(). In general, the left-hand side of (1) must be greater than or equalto the right-hand side of (1) for player 2 to accept player 1’s offer. However, (1) holdswith equality in equilibrium because of the following observation: If player 2 rejectsall offers, then it is optimal for player 1 to make no unilateral transfers. So player 2can guarantee a payoff of zero by rejecting all the offers. Therefore, the right-handside of (1) is greater than or equal to zero. The left-hand side is a continuous functionof x1, it takes a non-positive value at x1(z) = 0, and decreasing x1(z) increases player1’s payoff so that (1) holds with equality in equilibrium.

If player 2 makes the offer in period t, then

v − x2(z)

1 − δ1= −ψy2(z) + δ1

1 − δ1(v − x1(y2(z))) (2)

where player 2’s equilibrium strategy is to offer x2(z) given that player 2 consumedz in the previous period. The left-hand side of this equation is player 1’s payoff if heaccepts 2’s offer and the right-hand side is his payoff if he rejects it. In that case, player1 will make a unilateral transfer of y2(z), pay a one-period unilateral transfer cost ofψy2(z) and make an offer of x1(y2(z)) the next period, which will be accepted.

In general, the left-hand side of (2) must be greater than or equal to the right-handside of (2) for player 1 to accept player 2’s offer. Since player 1 can also ensure apayoff of zero for himself by making no unilateral transfers, making offers of zero andrejecting all offers, a similar continuity argument proves that (2) holds with equalityin equilibrium.

To simplify the expressions, multiply both sides of (1) by (1 − δ2) and multiplyboth sides of (2) by (1− δ1). I will use the following equations in the analysis. For allz,

x1(z) − (1 − δ2)φ [z − x1(z)]+ = (1 − δ2)(y1(z) − φ [z − y1(z)]+)

+ δ2(x2(y1(z)) − (1 − δ2)φ [y1(z) − x2(y1(z))]+

)

(3)

v − x2(z) = − (1 − δ1)ψy2(z) + δ1(v − x1(y2(z))) (4)

y1(z) and y2(z) are determined by

y1(z) = argmaxw

−(1 − δ1)ψw + δ1(v − x2(w)) (5)

andy2(z) = argmax

w−(1 − δ1)ψw + δ1(v − x1(w)) (6)

As I argued above, a strategy profile constitutes an equilibrium if and only if it satisfiesthese equalities. Therefore, in the rest of the appendix, I will solve for an equilibriumby solving the equation system (3), (4), (5) and (6).

123

492 B. Leventoglu

Proof of Proposition 1

Assume φ > φ∗. Suppose to the contrary that y1(z) = y2(z) = 0 in an equilibrium.Offers of such equilibrium coincide with x∗

1 and x∗2 , the unique equilibrium offers of

the standard Rubinstein game.Consider a subgame starting at a node at which z = 0, player 1 has just rejected

player 2’s offer and is about to make a unilateral transfer of y2(0) = 0. Player 1’scontinuation payoff from following the presumed equilibrium strategy at this node isδ1(v − x∗

1 ).

Alternatively, consider a one-period deviation with a unilateral transfer of y′2 =

ε > 0 for some small enough ε . Substitute in (3) z = ε, y1(z) = 0 and x2(y1(z)) =x∗2 and obtain

x1(ε) = −(1 − δ2)φε + δ2x∗2

Note that we choose ε small enough that the new solution leads to ε < x1(ε). Sincethe proposed equilibrium has offers of x∗

1 and x∗2 with y1(z) = y2(z) = 0, 0 < x∗

1 andEqs. (3) and (4) are continuous around z = 0 at such equilibrium, such ε exists. Alsonote that δ2x∗

2 = x∗1 .

Then player 1’s continuation payoff after such deviation becomes

−(1 − δ1)ψε + δ1(v − x1(ε))

which is equivalent to (after substituting x1(ε))

δ1(1 − δ2)(φ − φ∗) + δ1(v − x∗1 )

which is greater than δ1(v − x∗1 ) since φ > φ∗. That is, player 1 has a prof-

itable deviation from the presumed equilibrium strategy, a contradiction. This provesProposition 1.

Benchmark equilibrium analysis

I give the proof of Theorem 2 in this section. Since past consumption level of z doesnot appear in the objective function of the optimization problem for yi (z), neither ofy1(z) and y2(z) depends on z. Then the right- hand side of (4) becomes independentof z, so that x2(z) on the left-hand side also becomes independent of z. That is,

x2(z) = x2 all z

for some x2 ≥ 0. Therefore, the objective function of (5) is a decreasing function ofw. Then optimality implies that

123


y1(z) = 0

for all z.Substituting y1(z) = 0 in (3), we obtain

x1(z) − (1 − δ2)φ [z − x1(z)]+ = −(1 − δ2)φz + δ2x2(0)

so that

x1(z) ={

−(1 − δ2)φz + δ2 x2 if x1(z) ≥ zδ2 x2

1+(1−δ2)φif x1(z) ≤ z

(7)

Let z be such that

δ2 x21 + (1 − δ2)φ

= −(1 − δ2)φ z + δ2 x2

Then

z = δ2 x21 + (1 − δ2)φ

(8)

and x1(z) ≥ z if and only if z ≤ z. Also notice that x1(z) = z. Then x1(z) becomes

x1(z) ={

(1 + (1 − δ2)φ)z − (1 − δ2)φz if z ≤ zz if z ≥ z

(9)

Substituting this in the objective function of (6), we obtain

f (w) = −(1 − δ1)ψw + δ1(v − x1(w))

=⎧⎨

⎩

(−(1 − δ1)ψ + δ1(1 − δ2)φ)w

+ δ1(v − (1 + (1 − δ2)φ)z) if w ≤ z−(1 − δ1)ψw + δ1(v − z) if w ≥ z

Let

φ∗ = (1 − δ1)ψ

δ1(1 − δ2).

Then, if φ ≤ φ∗, f is a non-increasing function of w for w ≤ z and a decreasingfunction of w for w > z. In this case, optimality implies that y2(z) = 0. If φ > φ∗,f is a increasing in w for w ≤ z and decreasing for w > z. Then optimality impliesy2(z) = z for all z. In summary,

y2(z) ={0 if φ ≤ φ∗z if φ > φ∗ (10)

for all z.

123

494 B. Leventoglu

Case 1: φ ≤ φ∗In this case, y1(z) = y2(z) = 0. Rewriting (3) and (4) at z = 0, we obtain

x1(0) = δ2x2(0)

v − x2(0) = δ1(v − x1(0)

which yields x1(0) = x∗1 and x2(0) = x2(z) = x2 = x∗

2 so that

z = δ2x∗2

1 + (1 − δ2)φ

and

x1(z) ={

x∗1 − (1 − δ2)φz if z ≤ x∗

11+(1−δ2)φ

x∗1

1+(1−δ2)φif z ≥ x∗

11+(1−δ2)φ

Case 2: φ > φ∗In this case, y1(z) = 0 and y2(z) = z.Rewriting (3) and (4) at z = 0 and substituting

x1(z) = z, we obtain

x1(0) = δ2x2(0)

v − x2(0) = −(1 − δ1)ψ z + δ1(v − z)

Substitute x2(0) = x2 and z from (8) and obtain

z = δ2(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)φ − δ2(1 − δ1)ψ,

x2 = (1 + (1 − δ2)φ)(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)φ − δ2(1 − δ1)ψ

x1(z) ={

δ2 x2 − (1 − δ2)φz if z ≤ zδ2 x2

1+(1−δ2)φif z ≥ z

Substitute z∗ and z to obtain the expressions in Theorem 2. This completes the proofof Theorem 2.

Two-sided habit formation

I give the proof of Theorem 3 in this section. Let s be player 1’s income from otherresources. Let z denote player 1’s unilateral transfer the previous period, so that s −ψzis player 1’s consumption level and z is player 2’s consumption in the previous periodon a given off-the-equilibrium path. The condition for 1’s offer is the same as before:For all z,

123


x1(z) − (1−δ2)φ [z − x1(z)]+ = (1 − δ2)(y1(z)−φ [z − y1(z)]+)

+ δ2(x2(y1(z)) − (1 − δ2)φ [y1(z) − x2(y1(z))]+

)

(11)

Consider player 2’s offer x2(z). If 1 accepts the offer, hewill consume s and v−x2(z)forever, which will generate a lifetime payoff of s+v−x2(z)

1−δ1. Player 1’s previous period

consumption is s − ψz, which is less than his current consumption s + v − x2(z)since v ≥ x2(z), so that he does not pay any cost for habit. If he rejects the offer, hemakes a unilateral transfer of y2(z) to player 2, so his current consumption becomess − ψy2(z), for which he pays a one-period cost of γ [(s − ψz) − (s − ψy2(z))]+ =γψ [y2(z) − z]+ for his habit in addition to the cost of transfer ψy2(z). His offer of

x1(y2(z)) is accepted next period, which generates a lifetime payoff of s+v−x1(y2(z))1−δ1

from the next period on. Since s − ψy2(z) ≤ s + v − x1(y2(z)), he does not pay anyadditional cost the following period. Then the condition for 2’s offer becomes (aftermultiplying both sides by 1 − δ1)

s + v − x2(z) = (1 − δ1)((s − ψy2(z)) − γψ [y2(z) − z]+

)

+ δ1(s + v − x1(y2(z)))

equivalently

v − x2(z) = −(1 − δ1)(ψy2(z) + γψ [y2(z) − z]+

) + δ1(v − x1(y2(z))) (12)

After multiplying player 1’s payoff by 1 − δ1, y1(z) and y2(z) are determined by

y1(z) = argmaxw

s − (1 − δ1)(ψw + γψ [w − z]+) + δ1(v − x2(w)) (13)

and

y2(z) = argmaxw

s − (1 − δ1)(ψw + γψ [w − z]+) + δ1(v − x1(w)) (14)

where, s is player 1’s life time utility from consumption of s, ψw is a one-period costof unilateral transfer, γψ [w − z]+ is a one-period cost from habits and v − xi (w),

i = 1, 2, is his life time utility from agreement in the next period.Suppose that there is some z such that y2(z) < z. Then [y2(z) − z]+ = 0. This

implies that y2(z) also solves

argmaxw

s − (1 − δ1)ψw + δ1(v − x1(w))

which is independent of z. Since y2(z) < z′ for z′ ≥ z and so[y2(z) − z′]

+ = 0, wealso obtain y2(z′) = y2(z) for any z′ ≥ z. Let z be infimum of the set of all z’s suchthat y2(z) < z. Then for z > z, y2(z) = y2 for some y2 and [y2(z) − z]+ = 0. Fory2(z) < z′ to hold for all z′ ≥ z, it must be the case that y2 = z.

123

496 B. Leventoglu

Now consider z ≤ z. By choice of z, we have y2(z) ≥ z so that [y2(z) − z]+ =y2(z) − z for all z ≤ z. So y2(z) also solves

argmaxw

s − (1 − δ1)(ψw + γψ(w − z)) + δ1(v − x1(w))

subject to w ≥ z

Suppose that the solution is independent of z. Then the only possibility is y(z) = z forz ≤ z for the constraint for w ≥ z to be satisfied for all such z. Otherwise the solutionto this problem depends on z only if the constraint w ≥ z is binding, i.e., y(z) = z. Inthis case, [y2(z) − z]+ = 0 so that y2(z) also solves

argmaxw

s − (1 − δ1)ψw + δ1(v − x1(w))

because the constraint w ≥ z is relevant only to satisfy [y2(z) − z]+ = y2(z) − z but[y2(z) − z]+ = 0 when y(z) = z and [w − z]+ = 0 for all w ≤ z as well. But thenthe same logical steps lead to the same solution above, a contradiction with y(z) = z.Therefore, y(z) = z for all z.

Then (12) implies that x2(z) = x2 for some x2, all z. In turn, the objective functionof (13) becomes strictly decreasing inw so that y1(z) = 0 for all z ≥ z.Then followingthe same steps in Sect. 1, we obtain the same solution up to Eq. (9).


f (w) =⎧⎨

⎩

(−(1 − δ1)(1 + γ )ψ + δ1(1 − δ2)φ)w

+ (1 − δ1)γ z + δ1(v − (1 + (1 − δ2)φ)z) if w ≤ z−(1 − δ1)ψw + δ1(v − z) if w ≥ z

Let

φ∗ = (1 − δ1)(1 + γ )ψ

δ1(1 − δ2).


y2(z) ={0 if φ ≤ φ∗z if φ > φ∗ (15)

for all z.Case 1: φ ≤ φ∗

In this case, y1(z) = y2(z) = 0. Rewriting (11) and (12) at z = 0, we obtain

x1(0) = δ2x2(0)

v − x2(0) = δ1(v − x1(0)

123



2 so that

z = δ2x∗2

1 + (1 − δ2)φ

and

x1(z) ={

x∗1 − (1 − δ2)φz if z ≤ x∗

11+(1−δ2)φ

x∗1

1+(1−δ2)φif z ≥ x∗

11+(1−δ2)φ

Case 2: φ > φ∗In this case, y1(z) = 0 and y2(z) = z. Rewriting (11) and (12) at z = 0 and

substituting x1(z) = z, we obtain

x1(0) = δ2x2(0)

v − x2(0) = −(1 − δ1)(1 + γ )ψ z + δ1(v − z)

Substitute x2(0) = x2 and z to obtain

z = δ2(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)φ − δ2(1 − δ1)(1 + γ )ψ,

x2 = (1 + (1 − δ2)φ)(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)φ − δ2(1 − δ1)(1 + γ )ψ

x1(z) ={

δ2 x2 − (1 − δ2)φz if z ≤ zδ2 x2

1+(1−δ2)φif z ≥ z

This completes the proof of Theorem 3.

Joy from excess consumption

I give the proof of Theorem4 in this section. The condition for player 1’s offer becomes

x1(z) + (1 − δ2)(λ [x1(z) − z]+ − φ [z − x1(z)]+

)

= (1 − δ2)(y1(z) + λ [y1(z) − z]+ − φ [z − y1(z)]+)

+ δ2(x2(y1(z))+(1 − δ2)

(λ [x2(y1(z))−y1(z)]+ − φ [y1(z) − x2(y1(z))]+

))

(16)

and the optimality condition for x2(z), y1(z) and y2(z) are the same as in the bench-mark.

The proof follows the steps of the proof in Sect. 1. Since past consumption levelof z does not appear in the objective function of the optimization problem for yi (z),neither of y1(z) and y2(z) depends on z. Then the right-hand side of (4) becomes

123

498 B. Leventoglu

independent of z, so that x2(z) on the left-hand side also becomes independent of z.That is,

x2(z) = x2 all z

for some x2 ≥ 0. Therefore, the objective function of (5) is a decreasing function ofw. Then optimality implies that

y1(z) = 0

for all z.Substituting y1(z) = 0 in (16), we obtain

x1(z) + (1 − δ2)(λ [x1(z) − z]+ − φ [z − x1(z)]+

)

= −(1 − δ2)φz + (1 + (1 − δ2)λ)δ2x2(0)

so that

x1(z) ={

− (1−δ2)(φ−λ)1+(1−δ2)λ

z + δ2 x2 if x1(z) ≥ z1+(1−δ2)λ1+(1−δ2)φ

δ2 x2 if x1(z) ≤ z(17)

Let z be such that

(1 + (1 − δ2)λ) δ2 x21 + (1 − δ2)φ

= − (1 − δ2)(φ − λ)

1 + (1 − δ2)λz + δ2 x2

Then

z = 1 + (1 − δ2)λ

1 + (1 − δ2)φδ2 x2 (18)

and x1(z) ≥ z if and only if z ≤ z. Also notice that x1(z) = z. Then x1(z) becomes

x1(z) ={

�((1 + (1 − δ2)φ)z − (1 − δ2)φz

)if z ≤ z

z if z ≥ z(19)

where

� = 1

1 + (1 − δ2)λ


f (w) = −(1 − δ1)ψw + δ1(v − x1(w))

=⎧⎨

⎩

(−(1 − δ1)ψ + �δ1(1 − δ2)φ)w

+δ1(v − �(1 + (1 − δ2)φ)z) if w ≤ z−(1 − δ1)ψw + δ1(v − z) if w ≥ z

123


Let

φ∗ = (1 − δ1)ψ

�δ1(1 − δ2)= (1 + (1 − δ2)λ)

(1 − δ1)ψ

δ1(1 − δ2)


y2(z) ={0 if φ ≤ φ∗z if φ > φ∗

for all z.Case 1: φ ≤ φ∗

In this case, y1(z) = y2(z) = 0. Rewriting (16) and (4) at z = 0, we obtain

x1(0) = δ2x2(0)

v − x2(0) = δ1(v − x1(0)


2 so that

z = 1 + (1 − δ2)λ

1 + (1 − δ2)φδ2x∗

2 = 1 + (1 − δ2)λ

1 + (1 − δ2)φx∗1

and

x1(z) ={

x∗1 − (1 − δ2)φz if z ≤ 1+(1−δ2)λ

1+(1−δ2)φx∗1

1+(1−δ2)λ1+(1−δ2)φ

x∗1 if z ≥ 1+(1−δ2)λ

1+(1−δ2)φx∗1

Case 2: φ > φ∗In this case, y1(z) = 0 and y2(z) = z. Rewriting (16) and (4) at z = 0 and

substituting x1(z) = z, we obtain

x1(0) = δ2x2(0)

v − x2(0) = −(1 − δ1)ψ z + δ1(v − z)

Substitute x2(0) = x2 and z from (18) and obtain

z = δ2(1 + (1 − δ2)λ)(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)(φ − δ1δ2λ) − δ2(1 − δ1)ψ(1 + (1 − δ2)λ),

x2 = (1 + (1 − δ2)φ)(1 − δ1)v

(1 − δ1δ2) + (1 − δ2)(φ − δ1δ2λ) − δ2(1 − δ1)ψ(1 + (1 − δ2)λ)

123

500 B. Leventoglu

x1(z) ={

δ2 x2 − (1 − δ2)φz if z ≤ z1+(1−δ2)λ1+(1−δ2)φ

δ2 x2 if z ≥ z

This completes the proof of Theorem 4.

Mutual consent for unilateral transfers

Optimality conditions for offers x1(z) and x2(z) in (3) and (4) remain the same.However, for player 2 to accept a unilateral offer, accepting the offer must provide apayoff at least as high as the payoff from rejecting it. This introduces constraints forthe choice of unilateral transfer. y1(z) and y2(z) are determined by

y1(z) = argmaxw

−(1 − δ1)ψw + δ1(v − x2(w))

subject to

(1 − δ2) (w − φ[z − w]+) + δ2 (x2(w) − (1 − δ2)φ[w − x2(w)]+)

≥ −(1 − δ2)φz + δ2x2(0) (20)

and

y2(z) = argmaxw

−(1 − δ1)ψw + δ1(v − x1(w))

subject to

(1 − δ2) (w − φ[z − w]+) + δ2 (x1(w) − (1 − δ2)φ[w − x1(w)]+)

≥ −(1 − δ2)φz + δ2x1(0) (21)

where the constraint in each optimization problem ensures that player 2 accepts theunilateral transfer w.

First I will show that the constraint is binding at the optimum of (20). Let z∗ solvethe unconstrained problem

y1(z) = argmaxw

−(1 − δ1)ψw + δ1(v − x2(w))

If z∗ solves the constrained problem (20) at z, then it solves it at any z′ ≥ z since theconstraint for z∗ holds at z and does not change or becomes slack at z′.

Let z be the infimum of set of z’s such that z∗ solves the constrained problem (20)at z. Then y1(z) = z∗ for all z > z, and the constraint is binding for all z ≤ z bychoice of z.

Consider z ≤ z. Since the constraint is binding, we obtain

(1 − δ2)(y1(z) − φ [z − y1(z)]+)

+ δ2(x2(y1(z)) − (1 − δ2)φ [y1(z) − x2(y1(z))]+

)

= −(1 − δ2)φz + δ2x2(0)

123


Substituting this on the right-hand side of (3), we obtain

x1(z) − (1 − δ2)φ [z − x1(z)]+ = −(1 − δ2)φz + δ2x2(0)

equivalently

x1(z) ={

δ2x2(0) − (1 − δ2)φz if z ≤ x1(z)δ2x2(0)

1+(1−δ2)φif z ≥ x1(z)

Let z such that

δ2x2(0) − (1 − δ2)φ z = δ2x2(0)

1 + (1 − δ2)φ

Then

z = δ2x2(0)

1 + (1 − δ2)φ(22)

and z ≤ x1(z) if and only if z ≤ z for all z ≤ z.Now solve for y2(z) at z ≤ z. Substitute x1(z) in the objective function of (21) and

obtain

f (w) = −(1 − δ1)ψw + δ1(v − x1(w))

=⎧⎨

⎩

(−(1 − δ1)ψ + δ1(1 − δ2)φ)w

+ δ1(v − (1 + (1 − δ2)φ)z) if w ≤ z−(1 − δ1)ψw + δ1(v − z) if w ≥ z

Let

φ∗ = (1 − δ1)ψ

δ1(1 − δ2).

Then, if φ ≤ φ∗, f is a non-increasing function of w for w ≤ z and a decreasingfunction of w for w > z. In this case, optimality implies that y2(z) = 0. If φ > φ∗,f is increasing in w for w ≤ z and decreasing for w > z. Then optimality impliesy2(z) = z for all z. In summary,

y2(z) ={0 if φ ≤ φ∗z if φ > φ∗

for all z ≤ z.In both cases of φ, the right-hand side of (4) becomes independent of z, so that

x2(z) on the left-hand side also becomes independent of z. That is,

x2(z) = x2 all z ≤ z

123

502 B. Leventoglu

for some x2 ≥ 0. Therefore, the objective function of (5) is a decreasing function ofw, and the constraint is trivially satisfied by w = 0 at z ≤ z. Then, optimality impliesthat

y1(z) = 0

for all z ≤ z. Then y1(z) = z∗ = 0 for z ≥ z because otherwise y1(z) would havea discontinuous jump, contradicting that it is a continuous solution of a continuousoptimization problem.

Next, consider the constraint in the optimization problem (21). The right-hand sideof the inequality of (21) is decreasing in z ≥ 0 and the left-hand side is independentof z for z < w and decreasing in z for z ≥ w at the same rate as the right side. Nowconsider the equilibrium values of z, x1(z) and x2 in Theorem 2 (ii), which satisfies(22) and x1(z) = z. The inequality of (21) at z = 0 with these values and w = zbecomes

z ≥ δ2x1(0) = δ22 x2

equivalently

δ2(1 + (1 − δ2)φ) ≤ 1

which holds if player 2 is impatient enough, that is, δ2 is small enough. Then theinequality of (21) holds also for z ≥ 0, and the rest follows the proof of Theorem 2.

Incomplete information with alternating offers

I will compute separating and pooling Bayesian equilibria of the game.

Player 2 makes the first offer

In this case, if player 1 knows that φ = φl , then yi = 0 for i = 1, 2 and theequilibrium offers are given by

x∗1 = δ2(1 − δ1)


2 = 1 − δ1

1 − δ1δ2v

If he knows that φ = φh, then the equilibrium offers are given by

xh1 = δ2(1 − δ1)

(1 − δ2)(1 + φh)v and xh

2 = 1 + (1 − δ2)φh

(1 − δ2)(1 + φh)(1 − δ1)v

in this case, y1 = 0 and y2 = xh1 .

123


Separating equilibrium

Suppose that there is a separating equilibrium in which type φl makes an offer of xl

and type φh makes an offer of xh �= xl . If both offers are accepted immediately, thetype with the lower offer would benefit bymaking the offer of the other type. So player1 has to reject at least one of the offers with positive probability. Suppose he rejectsxl with probability αl and xh with probability αh .

After observing xτ , player 1 updates his belief to φ = φτ , τ ∈ {l, h}, the gameturns into a complete information game, and equilibrium offers and unilateral transfersare given as above in the continuation game.

Then type φh does not imitate type φl if and only if

(1 − αh)xh + αh((1 − δ2)y2 + δ2xh1 ) = (1 − αh)xh + αh xh

1 ≥ (1 − αl)xl + αlδ2x∗1

And type φl does not imitate type φh if and only if

(1 − αl)xl + αlδ2x∗1 ≥ (1 − αh)xh + αh((1 − δ2)y2 + δ2xh

1 ) = (1 − αh)xh + αh xh1

Note that I have substituted y2 = xh1 above. These two inequalities imply that

(1 − αl)xl + αlδ2x∗1 = (1 − αh)xh + αh xh

1 (23)

If αl ∈ (0, 1), then player 1 is indifferent between accepting and rejecting xl andit must be the case that

v − xl = δ1(v − x∗1 ) ⇒ xl = x∗

2

Then (23) becomes

(1 − αl)x∗2 + αlδ2x∗

1 = (1 − αh)xh + αh xh1 (24)

Player 1’s payoff from receiving the offer of xh is

(1 − αh)(v − xh) + αh[−(1 − δ1)y2 + δ1(v − xh1 )]

If xh satisfies

v − xh > −(1 − δ1)y2 + δ1(v − xh1 )

then αh = 0 is optimal for 1 so that xh is not optimal for type φh, because if she offersxh + ε and even if player 1 updates her beliefs to φ = φh after observing this offer, itis optimal for player 1 to accept this offer. So

v − xh = −(1 − δ1)y2 + δ1(v − xh1 ) (25)

123

504 B. Leventoglu

must be satisfied in such equilibrium. Then substituting y2 = xh1 ,

xh = (1 − δ1)v + xh1 = xh

2

Then (24) implies

(1 − αl)x∗2 + αlδ2x∗

1 = (1 − αh)xh2 + αh xh

1

which solves for αl as

αl = x∗2 − (1 − αh)xh

2 − αh xh1

x∗2 − δ2x∗

1(26)

Next I will check if αl ∈ [0, 1]. Since xh2 > xh

1 , αl achieves its minimum whenαh = 0, then αl becomes

αlmin = x∗

2 − xh2

x∗2 − δ2x∗

1(27)

αlmin ≥ 0 if and only if x∗

2 ≥ xh2 , which is equivalent to φ ≥ φ∗, which holds. So

αl ≥ 0.αlmin ≤ 1 if and only if δ2x∗

1 ≤ xh2 . This last inequality is equivalent to

(δ22 − 1 + δ1δ2)(1 − δ2)φ ≤ (1 − δ1δ2) − δ22(1 − δ2)

Since 1− δ1δ2 > 1− δ2 > δ22(1− δ2), the right hand is positive. If δ22 −1+ δ1δ2 < 0,

equivalently δ1 ≤ 1−δ22δ2

, then this inequality is satisfied for all values of φ. Otherwise,

it is satisfied if and only if φ ≤ φ where

φ = δ2(1 − δ1δ2)

(1 − δ2)(δ22 − 1 + δ1δ2)

− 1 > φ∗.

If δ1 ≤ 1−δ22δ2

or φ∗ < φ ≤ φ, then the following is a separating Bayesian equilib-

rium. Type φl offers x∗2 , which player 1 accepts with probability 1 − αl

min. Type φh

offers xh2 < x∗

2 , which player 1 accepts with probability 1. Player 2 rejects any otheroffer bigger than xh

2 , and accepts any other offer smaller than xh2 . If player 2 offers x∗

2 ,

then player 1 updates his belief to φ = φl , otherwise, he updates his belief to φ = φh

and the players play the equilibrium of the associated complete information games.The equilibrium payoff of both types of player 2 is xh

2 .

Next Iwill show thatαh > 0 cannot hold in such separating equilibrium if δ1 ≤ 1−δ22δ2

or φ∗ < φ ≤ φ. Suppose that αh > 0. If type φh offers xh2 − ε instead, for small

ε > 0, it is optimal for player 1 to accept her offer because (i) if he updates his beliefso that φ = φh, his continuation payoff from rejecting it is −(1− δ1)y2 + δ1(v − xh

1 ),

which is less than v − xh2 + ε by (25). Given that player 1 will accept the offer of

xh2 − ε with probability 1, type φh’s payoff from offering xh

2 − ε is xh2 − ε, and her

payoff from offering xh2 is

123


(1 − αh)xh2 + αh((1 − δ2)y2 + δ2xh

1 ) = (1 − αh)xh2 + αh xh

1

Since (1 − αh)xh2 + αh xh

1 < xh2 for αh > 0, there exists small enough ε such that

(1 − αh)xh2 + αh xh

1 < xh2 − ε, which contradicts with the optimality of offering xh

2 .

There is no such separating equilibrium with αh > 0 when δ1 ≤ 1−δ22δ2

or φ∗ < φ ≤ φ.

Pooling equilibrium

Suppose that δ22 − 1 + δ1δ2 > 0 and φ > φ. No αh and αl pair satisfies (26),because αl > 1 even when αh minimizes αl in (27). So no separating equilibriumwith αl ∈ (0, 1) exists in this case. Alternatively, consider αh ∈ (0, 1), then player 1is indifferent between accepting and rejecting xh and it must be the case that

v − xh = −(1 − δ1)y2 + δ1(v − xh1 ) ⇒ xh = xh

2

Then (23) becomes

(1 − αl)xl + αlδ2x∗1 = (1 − αh)xh

2 + αh xh1 (28)

Player 1’s payoff from receiving the offer of xl is

(1 − αl)(v − xl) + αlδ1(v − x∗1 )

Suppose that xl satisfies

(v − xl) < δ1(v − x∗1 )

then optimality for player 1 requires αl = 1, so (28) becomes

δ2x∗1 = (1 − αh)xh

2 + αh xh1

which implies

αh = xh2 − δ2x∗

1

xh2 − xh

1

< 0

because δ2x∗1 > xh

2 in this case. So (v − xl) ≥ δ1(v − x∗1 ) must hold.

If xl satisfies

(v − xl) > δ1(v − x∗1 )

then αl = 0 is optimal. Then from (28),

xl = (1 − αh)xh2 + αh xh

1

123

506 B. Leventoglu

which implies that xl < xh2 for αh > 0 since xh

2 > xh1 . If type φh offers xh

2 − ε

instead, for small ε > 0, it is optimal for player 1 to accept her offer because (i)if he updates his belief so that φ = φh, his continuation payoff from rejecting it is−(1 − δ1)y2 + δ1(v − xh

1 ), which is less than v − xh2 + ε, the payoff from accepting

the offer; (ii) if he updates his belief so that φ = φl , his continuation payoff fromrejecting it is δ1(v − x∗

1 ), which is less than v − xh2 + ε because δ2x∗

1 > xh2 in this

case and δ1 < 1. Given that player 1 will accept the offer of xh2 − ε with probability

1, type φh’s payoff from offering xh2 − ε is xh

2 − ε, and her payoff from offering xh2 is

(1 − αh)xh2 + αh((1 − δ2)y2 + δ2xh

1 ) = (1 − αh)xh2 + αh xh

1

Since (1 − αh)xh2 + αh xh

1 < xh2 for αh > 0, there exists small enough ε such that

(1 − αh)xh2 + αh xh

1 < xh2 − ε, which contradicts with the optimality of offering xh

2 .

So xl satisfies (v − xl) ≤ δ1(v − x∗1 ).

So xl must satisfy

(v − xl) = δ1(v − x∗1 )

which implies xl = x∗2 . Then from (28)

(1 − αl)x∗2 + αlδ2x∗

1 = (1 − αh)xh2 + αh xh

1

This inequality cannot hold since δ2x∗1 > xh

2 in this case and x∗2 > xh

2 > xh1 . So no

such separating equilibrium exists either.I will show that the following constitutes a pooling equilibrium in this case: Both

types of player 2 offer xh2 . Player 1 rejects offers x > xh

2 and accept offers x ≤ xh2 . If

player 2 offers x �= xh2 , then player 1 updates his belief to φ = φh and plays according

to the equilibrium of the associated complete information game.Since xh

2 is the lowest offer player 1 can secure in any equilibrium, it is optimal forhim to accept x ≤ xh

2 . Consider a continuation game after a rejection. In this case,player 1 believes that φ = φh and there will be no updating of his degenerate beliefs.So it is optimal for him to play his equilibrium strategy of the associated completeinformation game. Since there will be no updating in beliefs, it is optimal for bothtypes of player 2 to offer xh

2 and accept any offer x ≥ xh1 .

Given player 1’s strategy in the continuation game, both types of player 2 willachieve the payoff of −(1 − δ1)y2 + δ1(v − xh

1 ) = xh2 after a rejection. Given that

player 1 rejects any offer x > xh2 , then it is optimal for both types to offer xh

2 , whichplayer 1 accepts with probability 1. Since x > xh

2 is an off equilibrium offer, player 1can set his belief to φ = φh after receiving any offer of x > xh

2 .

The equilibrium payoff of both types of player 2 is xh2 in the separating and pooling

equilibria. Player 1’s payoff in the separating equilibrium is given by

θ(v − xh2 ) + (1 − θ)

[αlδ1(v − x∗

1 ) + (1 − αl)(v − x∗2 )

]

and his payoff in the pooling equilibrium is v − xh2 .

123


When player 1 makes the offer

x∗1 is the highest offer player 1 makes in any equilibrium. So both types of player2 accept any offer x ≥ x∗

1 . Also xh1 is the lowest offer that player 1 makes in any

equilibrium.Consider a separating equilibrium in which player 1 offers x ∈ [xh

1 , x∗1 ]. Since the

continuation game becomes a complete information game and δ2x∗2 = x∗

1 , type φl

rejects any offer x < x∗1 and offers x∗

2 the next period, which player 1 accepts. Thentype φh must be accepting x < x∗

1 in the separating equilibrium. But she can achieveδ2x∗

2 = x∗1 > x by imitating type φl . So x < x∗

1 cannot be a separating equilibriumoffer. Since both types accept x∗

1 , x∗1 cannot be a separating equilibrium offer either.

So no separating equilibrium exists.Consider a pooling equilibrium in which player 1 offers x ∈ [xh

1 , x∗1 ]. Suppose that

type φτ rejects x with probability βτ , τ ∈ {l, h}. After a rejection, player 1 updateshis beliefs according to Bayes’ rule to

θ ′ = Pr(φ = φh) = θβh

θβh + (1 − θ)βl.

Suppose that βh > 0. Then θ ′ ∈ (0, 1). Since player 2 makes the offer the next period,the continuation game has either a separating equilibrium or a pooling equilibrium. Inboth cases, the payoff of both types of player 2 is equal to xh

2 (see the conclusion ofthe previous section). Then both types accept any offer x ≥ δ2xh

2 in the first period. Soplayer 1 offers δ2xh

2 in such a equilibrium. Since player 2 of both types is indifferentbetween accepting and rejecting the offer, they can reject it with any probability. Sincerejection is costly for player 1, player 1 can avoid rejection by offering δ2xh

2 + ε. Forδ2xh

2 to be optimal, it must be that βh = βl = 0. Since any rejection is off equilibriumin this case, player 1 can set his belief to φ = φh . Given that he will believe thatφ = φh after a rejection, the game will turn into a complete information game withφ = φh, so both types of player 2 can achieve at most xh

2 in the continuation gameafter rejection. So it is optimal for them to accept player 1’s offer of δ2xh

2 in the firstperiod.

In this case, the information rent type φh collects is

R(φh) = δ2xh2 − xh

1 = δ2(1 − δ2)(1 − δ1)φh

(1 − δ2)(1 + φh)v > 0

and

R′(φh) = R(φh)

1 + φh> 0

and(

R(φh)

xh1

)′= ((1 − δ2)φh)′ = 1 − δ2 > 0

123

508 B. Leventoglu

References

Abel, A.B.: Asset prices under habit formation and catching up with the joneses. Am. Econ. Rev. 80, 38–42(1990)

Apesteguia, J., Ballester,M.A.:A theory of reference-dependent behavior. Econ. Theory 40, 427–455 (2009)Avery, C., Zemsky, P.B.: Money burning and multiple equilibria in bargaining. Game Econ. Behav. 7,

154–168 (1994)Busch, L.A., Wen, Q.: Perfect equilibria in a negotiation model. Econometrica 63, 545–565 (1995)Berejikian, J.D.: International Relations under Risk: Framing State Choice. State University of New York

Press, Albany (2004)Braun, P.A., Constantinides, G.M., Ferson, W.E.: Time nonseparability of aggregate consumption: interna-

tional evidence. Eur. Econ. Rev. 37, 897–920 (1993)Campbell, J.Y., Cochrane, J.H.: By force of habit: a consumption-based explanation of aggregate stock

market behavior. J. Polit. Econ. 107, 205–251 (1999)Carroll, C.D., Overland, J., Weil, D.N.: Comparison utility in a growth model. J. Econ. Growth 2, 339–367

(1997)Camerer, C., Loewenstein, G.: Behavioral economics: past, present, and future. In: Camerer, C., Loewen-

stein, G., Rabin, M. (eds.) Advances in Behavioral Economics, pp. 3–51. princeton University Press,Princeton (2004)

Constantinides, G.M.: Habit formation: a resolution of the equity premium puzzle. J. Polit. Econ. 98,519–543 (1990)

Fernandez, R., Glazer, J.: Striking for a bargain between two completely informed agents. Am. Econ. Rev.81(1), 240–252 (1991)

Ferson, W.E., Constantinides, G.M.: Habit persistence and durability in aggregate consumption: empiricaltests. J. Financ. Econ. 29, 199–240 (1991)

Fuhrer, J.C.: Habit formation in consumption and its implications for monetary-policy models. Am. Econ.Rev. 90, 367–390 (2000)

Haller, H.H., Holden, S.: A letter to the editor on wage bargaining. J. Econ. Theory 52, 232–236 (1990)Jonge Oudraat, C.: Making economic sanctions work. Survival 42, 105–127 (2000)Kahneman, D., Knetsch, J.L., Thaler, R.H.: The endowment effect, loss aversion, and status quo bias. J.

Econ. Perspect. 5, 193–206 (1991)Kahneman, D., Tversky, A.: Prospect theory: an analysis of decisions under risk. Econometrica 47, 313–327

(1979)Levy, J.S.: Loss aversion, framing and bargaining: the implications of prospect theory for international

conflict. Int. Polit. Sci. Rev. 17, 179–195 (1996)Levy, J.S.: Prospect theory and the cognitive-rational debate. In: Geva, N., Mintz, A. (eds.) Decisionmaking

on War and Peace: The Cognitive-Rational Debate, pp. 33–50. Lynne Rienner, Boulder (1997a)Levy, J.S.: Prospect theory, rational choice, and international relations. Int. Stud. Quart. 41, 87–112 (1997b)Merlo, A., Wilson, C.: Efficient delays in a stochastic model of bargaining. Econ. Theory 11, 39–55 (1998)Muthoo, A.: A note on repeated-offers bargaining with one-sided incomplete information. Econ. Theory 4,

295–301 (1994)Orrego, F.: Habit formation and indeterminacy in overlapping generations models. Econ. Theory 55:225–

241 (2014)Osborne, M.J., Rubinstein, A.: Bargaining and Markets. Academic Press, London (1990)Ponsatí, C., Sákovics, J.: Rubinstein bargaining with two-sided outside options. Econ. Theory 11, 667–672

(1998)Rozen, K.: Foundations of intrinsic habit formation. Econometrica 78, 1341–1373 (2010)Rubinstein, A.: Perfect equilibrium in a bargaining model. Econometrica 50, 97–110 (1982)

123

Bargaining with habit formation - Duke...

Documents

Transcript of Bargaining with habit formation - Duke...